Case \(\mathfrak {m}(X)<\infty \) .
Since \(X\)  admits a superlinear isoperimetric profile, the fact that the embedding of \(W^{1,p}(X,\mathsf {d},\mathfrak {m})\)  in \(L^p(X,\mathfrak {m})\)  is compact is a direct consequence of Proposition REF .
We can thus appeal to the standard spectral theory (see e.g. [1]}) to infer that \(H_t\)  is a compact operator for every \(t>0\)  and that \(\sigma (-\Delta )\)  consists of a sequence of isolated eigenvalues \((\lambda _{k})_{k\in \mathbb {N}}\subset [0,\infty )\)  with \(\lambda _{k}\rightarrow \infty \)  as \(k\rightarrow \infty \)  and finite dimensional associated eigenspaces.
We can also appeal to the variational characterisation of the eigenvalues (see [2]} to infer that
\(\lambda _{k}=\min _{S_{k+1}} \,\max _{f\in S_{k+1}, \Vert f\Vert _{2}=1} {2\mathsf {Ch}}_{\mathfrak {m}}(f),\) 
where \(S_{k}\subset W^{1,2}(X,\mathsf {d}, \mathfrak {m})\)  denotes an arbitrary \(k\) -dimensional subspace.
Since \(f\equiv 1\)  is an element of \(L^{2}(X,\mathfrak {m})\) , we infer that \(\lambda _0=0\) . Moreover, from the Sobolev-to-Lipschitz property satisfied by \(\mathsf {RCD}(K,\infty )\)  spaces [3]}, it holds that \(\mathsf {Ch}_{\mathfrak {m},2}(f)=0\)  if and only if \(f\)  is constant \(\mathfrak {m}\) -a.e..
Thus \(\lambda _{0}=0\)  if and only if \(\mathfrak {m}(X)<\infty \)  and, in this case, \(f\)  is a \(\lambda _{0}\) -eigenfunction if and only if \(f\)  is constant \(\mathfrak {m}\) -a.e..
Specialising (REF ) to \(k=1\)  in case \(\mathfrak {m}(X)<\infty \)  gives the claimed variational formula for \(\lambda _{1}\)  which trivially coincides with (REF ) by the very definition of Cheeger energy.
We finally prove the implication “if \(f\)  satisfies (REF ) then \(f\)  is a \(\lambda _{1}\) -eigenfunction”.
We first show that \(\lambda _1 f\in \partial \mathsf {Ch}_{\mathfrak {m},2}(f)\) : for every \(g\in L^2(X,\mathfrak {m})\)  with \(\int _{X} g \, \mathsf {d}\mathfrak {m}=0\)  we have
\(\mathsf {Ch}_{\mathfrak {m},2}(g)\ge \frac{\lambda _1}{2}\int _X g^2\,\mathsf {d}\mathfrak {m}\ge \lambda _1\int _X fg\,\mathsf {d}\mathfrak {m}-\frac{\lambda _1}{2}\int _X f^2\mathsf {d}\mathfrak {m}=\mathsf {Ch}_{\mathfrak {m},2}(f)+\int _X \lambda _1f(g-f)\mathsf {d}\mathfrak {m}\) 
as a consequence of the variational characterization of \(\lambda _{1}\) , a trivial inequality and (REF ).
The function \(\lambda _1f\)  is also the element of minimal norm in the set \(\partial \mathsf {Ch}_{\mathfrak {m},2}(f)\) , and thus \(-\Delta f=\lambda _1f\) , since
\(\Vert  \Delta f\Vert _{2}\ge \int _X -f\Delta f\,\mathsf {d}\mathfrak {m}=\int _X |\nabla f|^2\,\mathsf {d}\mathfrak {m}=\lambda _1=\Vert  \lambda _1 f\Vert _{L^2}\, ,\) 
where we have used again (REF ) and the Cauchy-Schwarz inequality.

Case \(\mathfrak {m}(X)=\infty \) .
The assumption on the spectrum implies the existence of the first eigenvalue \(\lambda _0\)  of \(-\Delta \)  which is isolated and with non-empty finite dimensional eigenspace. Using the Sobolev-to-Lipschitz property satisfied by \(\mathsf {RCD}(K,\infty )\)  spaces [3]} and observing that the constant functions are not in \(L^2(X,\mathfrak {m})\) , we have \(\lambda _0>0\) .
The variational characterization of \(\lambda _{0}\)  still holds (see in this case [2]}) and gives
\(\lambda _0=\min _{\Vert f\Vert _{2}=1}\Big \lbrace 2\mathsf {Ch}_{\mathfrak {m},2}(f)\Big \rbrace ,\) 
so that \(\lambda _0\)  trivially corresponds to the definition given in (REF ) by the very definition of Cheeger energy. It remains to prove that a function \(f\)  such that
\(\int _X f^2\mathsf {d}\mathfrak {m}=1 \ \ \textrm {and} \ \ \int _X |\nabla f|^2\mathsf {d}\mathfrak {m}=\lambda _0\) 
is a \(\lambda _0\) -eigenfunction.
We first show that \(\lambda _0 f\in \partial \mathsf {Ch}_{\mathfrak {m},2}(f)\) : for every \(g\in L^2(X,\mathfrak {m})\)  we have
\(\mathsf {Ch}_{\mathfrak {m},2}(g)\ge \frac{\lambda _0}{2}\int _X g^2\,\mathsf {d}\mathfrak {m}\ge \lambda _0\int _X fg\,\mathsf {d}\mathfrak {m}-\frac{\lambda _0}{2}\int _X f^2\mathsf {d}\mathfrak {m}=\mathsf {Ch}_{\mathfrak {m},2}(f)+\int _X \lambda _0f(g-f)\mathsf {d}\mathfrak {m}\) 
as a consequence of the variational characterization of \(\lambda _{0}\) , a trivial inequality and (REF ).
The function \(\lambda _0f\)  is also the element of minimal norm in the set \(\partial \mathsf {Ch}_{\mathfrak {m},2}(f)\) , and thus \(-\Delta f=\lambda _0f\) , since
\(\Vert  \Delta f\Vert _{2}\ge \int _X -f\Delta f\,\mathsf {d}\mathfrak {m}=\int _X |\nabla f|^2\,\mathsf {d}\mathfrak {m}=\lambda _0=\Vert  \lambda _0 f\Vert _{L^2}\, ,\) 
where we have used again (REF ) and the Cauchy-Schwarz inequality.
